A particle which is constrained to move along the $x-$axis, is subjected to a force in the same direction which varies with the distance $x$ of the particle from the origin as $F(x) = - kx + a{x^3}$. Here k and a are positive constants. For $x \ge 0$, the functional from of the potential energy $U(x)$ of the particle is
A particle which is constrained to move along the $x-$axis, is subjected to a force in the same direction which varies with the distance $x$ of the particle from the origin as $F(x) = - kx + a{x^3}$. Here k and a are positive constants. For $x \ge 0$, the functional from of the potential energy $U(x)$ of the particle is
A particle which is constrained to move along the $x-$axis, is subjected to a force in the same direction which varies with the distance $x$ of the particle from the origin as $F(x) = - kx + a{x^3}$. Here k and a are positive constants. For $x \ge 0$, the functional from of the potential energy $U(x)$ of the particle is
$F = \frac{{ - dU}}{{dx}} \Rightarrow dU = - F\,\,dx$
$ \Rightarrow U = - \int_0^x {( - Kx\, + \,a{x^3})dx} $$ = \frac{{k{x^2}}}{2} - \frac{{a{x^4}}}{4}$
$\therefore $ We get $U = 0$ at $x = 0 $ and $x =$ $\sqrt {2k/a} $
and also $U =$ negative for $x > \sqrt {2k/a} $.
So $F = 0$ at $x = 0$
i.e. slope of $U -x$ graph is zero at $x = 0.$
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